Two More Restricted Versions of the QAP

Abstract

In this chapter which is a continuation of the previous one, we consider two restricted version of the QAP: the Anti-Monge—Toeplitz QAP and the KalmansonToeplitz QAP. The Anti-Monge—Toeplitz QAP is a restricted version of the QAP, where one of the coefficient matrices is a left-higher graded Anti-Monge matrix and the other one is a symmetric Toeplitz matrix. The interest in this problem is motivated by a number of practical applications, e.g. the turbine problem and the data arrangement problem, some of which will be considered in detail in the second section of this chapter. Moreover, the Anti-Monge—Toeplitz QAP contains the TSP on symmetric Monge matrices as a special case. Despite the very special structure of the Anti-Monge—Toeplitz QAP, i.e., the quite restrictive conditions to be fulfilled by its coefficient matrices, the problem is NP-hard. Namely, the turbine problem which is a special case of the Anti-Monge—Toeplitz QAP is NPhard, as shown by Burkard, Çela, Rote and Woeginger [33] . However, for Toeplitz matrices satisfying some additional conditions, (e.g. benevolent, k-benevolent, or bandwidth-2 matrices) , the Anti-Monge Toeplitz QAP becomes polynomially solvable. These polynomially solvables special cases of the problem which are constant permutation QAPs, will be described in detail in the first section of this chapter. Then, a polynomially solvable version of the Kalmanson-Toeplitz QAP, which i.e. a QAP with a Kialmanson and a Toeplitz matrix, is described. Further, so-called permuted polynomially solvable cases of the QAP are briefly discussed.